Wednesday, December 14, 2011

American football fans are abuzz with what has been termed "Tebow-mania" or "Tebow Fever." The reference is to Denver Broncos' quarterback Tim Tebow, whose unorthodox playing style, his "mixing faith with football" (in the words of one writer), and his team's winning ways have garnered him great attention.

As seen in the Broncos' game-by-game log, Denver won in Tebow's first start, October 23 at Miami, lost the next week to Detroit, and then went on a still-ongoing six-game winning streak. Amanda Rykoff, writing in an ESPN-W "roundtable discussion" captures the essence of Tebow-mania:

...I'm not going to try to explain why he's 7-1 as a starter for the Broncos this year. Or why Denver has been able to come from behind late in the fourth quarter in five of those seven victories. Or how the Broncos managed to win what seemed to be an absolutely unwinnable game on Sunday against the Bears, trailing 10-0 with two minutes left in regulation.

The blog "This Given Sunday" has a detailed summary of each Broncos/Tebow win during the stretch.

The improbable nature of many Bronco wins has led probability analysts to estimate the likelihood of Denver's comeback-heavy winning streak. The key element is the win probability in each game, based on historical data. If you click here to get to the Advanced NFL Stats website and then scroll down to the graph with all the wavy lines, you'll get an idea of win probability. First, you would select one of the lines, corresponding to how your team is doing with regard to score (e.g., leading by 7, trailing by 3). Once you have your line, then follow it along the horizontal axis corresponding to how many minutes remained in the game. For example, a team trailing by 7 entering the fourth quarter tends to win about 10% of the time.

Using a chart like the one above, analysts would then find the win probabilities for Denver in each of its victories, given the depth of Denver's dire circumstances in a given game (i.e., when it trailed the most with the least time remaining). One can then multiply the probabilities together to get an overall probability estimate of the Broncos' streak. (This is analogous to calculating the probability of rolling double-sixes with a pair of dice by multiplying 1/6, which is the probability of a six on one die, by 1/6, the probability of a six on the other die, to yield 1/36.)

Using this methodology, ESPN's Statistics and Information Blog estimates a probability of Denver winning its last six straight at "approximately one in 137,000. The odds are better that a flipped coin comes up heads 17 consecutive times."

"This Given Sunday" (the blog cited above) calculates the probability of the Broncos winning the seven games with Tebow as starter (ignoring the one loss). The verdict: "the odds of the Broncos winning all seven games from their lowest odds in each particular game situation [are] 1 in 27 million."

If you're going to use seven as the win total, you must take into account the one loss, in my view. The seven game-specific probabilities from "This Given Sunday" are as follows:

.01, .15, .58, .17, .18, .14, .01

If you throw out the high and low values, the Tebow-led Broncos typically faced around a .15 probability of winning in many of the games during the stretch. Using an online tool known as a Binomial Probability Calculator, we can ask the question: For a team that faced only a .15 probability of winning each game, what is the likelihood of that team winning seven or more out of eight games? The answer, given these assumptions, is .00001 or 1-in-100,000.

As many writers have acknowledged, Tebow obviously should not get sole credit for Denver's winning stretch. However, his contribution appears to be great. One metric is quarterback efficiency ratings, which attempt to boil down many passing statistics (e.g., completion rate, yardage gained, touchdowns, interceptions) into a single number.

ESPN's Statistics and Information Blog notes that Tebow has the highest score (96.3) on one such metric, the Total QBR, of all NFL quarterbacks this season in the final 7 minutes of the fourth quarter (using a certain minute-mark rather than, say, the fourth quarter as a whole seems a little arbitrary; would Tebow still lead if we used the final 8 minutes or 6 minutes?).

Along with Denver's defense, which has had to shut down opposing offenses, another key figure in the Broncos' recent success is kicker Matt Prater. Without his hot foot, the winning streak would be over. According to Adena Andrews's commentary in the aforementioned ESPN-W roundtable:

Prater, who was recognized as the AFC special teams player of the week after his performance against the Bears on Sunday (a 59-yard field goal to tie the game, and then a 51-yarder for the overtime victory), has hit 28 of the 29 career field goals he has attempted in the fourth quarter or overtime.

One final factor to consider is that, as unusual as the Broncos' stretch appears, maybe in the larger historical scheme it is not so unexpected. The National Football League has been around for roughly 90 years. Initially, the league had around 10 teams, and increased over the years to the teens and mid-twenties (with the NFL-AFL merger), and continued to expand to the present 32 teams. The number of games per team per season has increased from roughly 10-12 in the early years to the present 16.

Let's say that, in a given season under the modern schedule, each team would have 10 opportunities to begin a winning stretch for six games (e.g., right from the start, beginning after Week 1, beginning after Week 2, etc.). Once Week 11 had gone by, of course, it would no longer be possible to start a six-game winning streak.

As a simplification, let's say further that within each of the most recent 30 seasons, there were 300 opportunities for a six-game winning streak (roughly 30 teams X 10 opportunities); that for each of the prior 30 years, there were 200 opportunities; and for the first 30 years of the NFL, there were 100 annual opportunities. That yields roughly 18,000 opportunities. Considering the above likelihood estimate of the Broncos' winning six straight in the comeback fashion they did (i.e., 1-in-137,000), it seems that Denver's recent feat goes beyond the ordinary course of events.